\[\begin{equation}
P_W = (2G_B - G) \frac{\mu}{m}.
\end{equation}\] where \(G_B\) is the Gini index of inequality between the two halves, \(G\) is the Gini index for the entire distribution, \(\mu\) is the mean of the distribution, and \(m\) is the median. Letting \(L(p)\) be the ordinate of the Lorenz curve at the \(p_{th}\) percentile, the discrepancy between perfect equality and the Lorenz curve at the median is: \(D_{50} = 0.5 - L(0.5)\). Given that \(G_B = D_{50}\), the Gini index can be rewritten as \(G=G_B + G_W\) with \(G_W\) representing inequality within the two halves of the distribution. This implies the polarization index is: \[\begin{equation}
P_W = (G_B-G_W)\frac{\mu}{m}.
\end{equation}\]
Bipolarization
Bipolarization
Components
Alienation: Increasing gap between the two halves
Group Identification: decrease in local variance
Bipolarization vs. Inequality
Not synonyms
Pigou-Dalton: progressive income transfer must reduce inequality
Same transfer may increase polarization (e.g., Musk to Trump)
The parameter \(\alpha\) is the degree of sensitivity to polarization. When \(\alpha=0\), \(P_{ER}(\alpha) = 2 G_{B}\), where \(G_B\) is the Gini index between groups.
\[
I(y) = \sum_{g}^K w_g I_g + I(\mu_1 e_1, \mu_2 e_2, \ldots, \mu_k e_k)
\]\[
w_g =
\begin{cases}
f_g \left( \frac{\mu_g}{\mu} \right)^c & \text{if } c \neq 0, 1 \\
f_g \left( \frac{\mu_g}{\mu} \right) & \text{if } c = 1 \\
f_g & \text{if } c = 0.
\end{cases}
\]\(I_g\) is the inequality in group \(g\), \(\mu_g\) is the mean of incomes for members of group \(g\) and \(e_g\) is a unity vector with \(n_g\) elements, and \(f_g=\frac{n_g}{n}\). The parameter \(c\) measures transfer sensitivity. For \(c < 2\), the measure is more sensitive to transfers in the bottom of the distribution than to those in the top end of the distribution. When \(c=1\) or \(c=0\) the Theil (1972) measure applies.
Spatial Polarization
The measure of spatial polarization is the ratio of the between-group inequality over the within-group inequality: \[
P_{ZK} = \frac{ I(\mu_1 e_1, \mu_2 e_2, \ldots, \mu_k e_k)}{\sum_{g}^K w_g I_g }.
\]
The polarization can thus grow with increasing distances between the average incomes of the regions as well as a decrease in the inequality within the regions.
heirarchical scales of polarization (global-meso-local)
Dynamics
relax fixed regionalization assumption
extend traditional interregional inequality decomposition framework
Framework
Polarization as Graph Intersection
Areas as nodes
Each area will be a vertex \(v_i\), \(i=1,2,\ldots, n\)
\(|V| = n\)
Two Graphs
Distributional (Attribute) Polarization Graph
Geographical (Spatial) Graph
Attribute Polarization Graph
\[A = (V_A, E_A)\]
\[E_A = \{(i, j) \in V \times V \mid g(f(i)) = g(f(j))\}\] where \(g(f(i))\) is a function mapping \(i\) to a value class. The specific form of this mapping obtains from the type of polarization under consideration.
Geographical Adjacency Graph
\[G = (V_G, E_G)\] codifies the geographical relationships between each pair of locations.
Components
Both graphs share identical vertex sets, \(V_A = V_G = V\), with each vertex linked to a locational observation \(i\), and \(|V| = n\).
Let \(C_G = \{C_{G,1}, C_{G,2}, ..., C_{G,k_{G}}\}\) define the \(k_G\) connected components of \(G\).
Similarly, define \(C_A = \{C_{A,1}, C_{A,2}, ..., C_{A,k_{A}}\}\) as the component set for \(A\).
Let \(k=max(k_G, k_A)\).
Intersection Graph
\[I=(V_I, E_I)\] where
\(V_I=V_G=V_A\)
\(E_I = E_G \cap E_A\)
Define \(C_I = \{C_{I,1}, C_{I,2}, \ldots, C_{I,k_{I}}\}\) as the component set for \(I\), with \(k_{I} = \left | C_I \right |\).
Spatial Polarization Index
The index of spatial polarization is: \[
S(G, A) = 1 - \frac{k_I - k}{ (n - k)}
\]
\(k =max(k_G, k_A)\).
Spatial Bipolarization
Given a set of \(n\) locations with a vector of per capita incomes \(Y\), let \(m= \text{median}(Y)\). Define \(A\) as the \(n \times n\) attribute adjacency graph with edge conditions: \[
A_{ij} = \begin{cases}
1 & \text{if } (Y_i \leq m \text{ and } Y_j \leq m) \text{ or } (Y_i > m \text{ and } Y_j >m), \\
0 & \text{otherwise}.
\end{cases}
\]
\(A\) will have \(k_A=2\) components of equal size when \(n\) is even, and approximately equal size when \(n\) is odd.
Define \(G\) as the spatial adjacency graph with: \[
G_{ij} = \begin{cases}
1 & \text{if observations } i \text{ and } j \text{ are geographically adjacent}, \\
0 & \text{otherwise}.
\end{cases}
\] The number of connected components for the spatial adjacency graph, \(k_G\), depends on the locational configuration. In a connected spatial adjacency graph, there is a path between each pair of locations, and thus there will be no isolated vertices or disconnected subgraphs. In such a graph \(k_G=1\).
Spatial Bipolarization
\[\begin{equation}
S^B(A, G) = 1 - \frac{k_I-k}{n-2}.
\label{eq:sp}
\end{equation}\] with \(k_I = |C_I|\) , \(I = A \cap G\), and \(k = max(k_A, k_G)\).
For the nodes in this intersection component, this represents the proportion of incident edges from the geographical adjacency graph that also exist in the intersection component. In other words, this is the share of each location’s geographical neighbors that belong to the same attribute component as the location.
In a quartile-based grouping there would be four components in the attribute graph and, assuming the spatial graph has a single component, our index of spatial multipolarization becomes: \[
S^M(A, G) = 1 - \frac{k_I-4}{n-4}.
\]
with \(k_I = |C_I|\) , \(I = A \cap G\), and \(k = max(k_A, k_G)\).
Dynamics of Inequality and Spatial Polarization
Scales of Spatial Polarization
\[\begin{equation}
1 - \frac{k_{i,low} - 1}{n_{low}-1} = 1- \frac{1_{} - 1}{16-1} = 1
\end{equation}\] and for the high end of the distribution we have: \[\begin{equation}
1- \frac{k_{i,high} - 1}{n_{high}-1} = 1 - \frac{5 - 1}{16-1} = 0.73
\end{equation}\] Note that the overall spatial polarization index is the average of these two components.
Local neighborhood homogeneity index Mexico state income 1940.
Local class connectivity index Mexico state income 1940.
Spatial multipolarization graph 1940 (quartiles: Q1 dark blue, Q2 blue, Q3 red, Q4 dark red).
Esteban, J. M, and D. Ray. 1994. “On the Measurement of Polarization.”Econometrica 62 (4): 819–51.
Hanson, Gordon H. 1996. U.S.-Mexico Integration and Regional Economies. Cambridge: National Bureau of Economic Research.
Kaplan, Ethan, Jörg L. Spenkuch, and Rebecca Sullivan. 2022. “Partisan Spatial Sorting in the United States: A Theoretical and Empirical Overview.”Journal of Public Economics 211 (July): 104668. https://doi.org/10.1016/j.jpubeco.2022.104668.
Pike, Andy, Vincent Béal, Nicolas Cauchi-Duval, Rachel Franklin, Nadir Kinossian, Thilo Lang, Tim Leibert, et al. 2024. “‘Left Behind Places’: A Geographical Etymology.”Regional Studies 58 (6): 1167–79. https://doi.org/10.1080/00343404.2023.2167972.
Quah, Danny T. 1996. “Twin Peaks: Growth and Convergence in Models of Distribution Dynamics.”The Economic Journal 106 (437): 1045–55. https://doi.org/10.2307/2235377.
Rey, Sergio J., and Esau Casimiro Vieyra. 2023. “Spatial Inequality and Place Mobility in Mexico: 2000–2015.”Applied Geography 152 (March): 102871. https://doi.org/10.1016/j.apgeog.2023.102871.
Rodríguez-Pose, Andrés. 2018. “The Revenge of the Places That Don’t Matter (and What to Do about It).”Cambridge Journal of Regions, Economy and Society 11 (1): 189–209. https://doi.org/10.1093/cjres/rsx024.
Theil, Henri. 1972. Statistical Decomposition Analysis: With Applications in the Social and Administrative Sciences. Studies in Mathematical and Managerial Economics 14. Amsterdam: North-Holland Publ. [u.a.].
Zhang, Xiaobo, and Ravi Kanbur. 2001. “What Difference Do Polarisation Measures Make?: An Application to China.”Journal of Development Studies 37: 85–98.